1.6. Inverse conjecture over the integers 103

Exercise 1.6.17. Show that a degree ≤ 1 nilsequence of complexity M is

Fourier-measurable with growth function FM depending only on M, where

Fourier measurability was defined in Section 1.2.

Exercise 1.6.18. Show that the class of nilsequences of degree ≤ s does

not change if we drop the condition G = G≤0, or if we add the additional

condition G = G≤1.

Remark 1.6.11. The space of nilsequences is also unchanged if one insists

that the polynomial orbit be linear, and that the filtration be the lower cen-

tral series filtration; and this is in fact the original definition of a nilsequence.

The proof of this equivalence is a little tricky, though; see [GrTaZi2010b].

1.6.3. Connection with the Gowers norms. We define the Gowers

norm f

Ud[N]

of a function f : [N] → C by the formula

f

Ud[N]

:= f

Ud(Z/N Z)

/ 1[N]

Ud(Z/N Z)

where N is any integer greater than (d+1)N, [N] is embedded inside Z/N Z,

and f is extended by zero outside of [N]. It is easy to see that this definition

is independent of the choice of N . Note also that the normalisation factor

1[N]

Ud(Z/N Z)

is comparable to 1 when d is fixed and N is comparable to

N.

One of the main reasons why nilsequences are relevant to the theory of

the Gowers norms is that they are an obstruction to that norm being small.

More precisely, we have

Theorem 1.6.12 (Converse to the inverse conjecture for the Gowers norms).

Let f : [N] → C be such that f

L∞[N]

≤ 1 and |f, ψ L2([N])| ≥ δ for some

degree ≤ s nilsequence of complexity at most M. Then f

Us+1[N]

s,δ,M

1.

We now prove this theorem, using an argument from [GrTaZi2009]. It

is convenient to introduce a few more notions. Define a vertical character

of a degree ≤ s filtered nilmanifold G/Γ to be a continuous homomorphism

η : G≥s → R/Z that annihilates Γ≥s, or equivalently an element of the

Pontryagin dual G≥s/Γ≥s of the torus G≥s/Γ≥s. A function F : G/Γ → C

is said to have vertical frequency η if F obeys the equation

F (gsx) = e(η(gs))F (x)

for all gs ∈ G≥s and x ∈ G/Γ. A degree ≤ s nilsequence is said to have

a vertical frequency if it can be represented in the form n → F (O(n)) for

some Lipschitz F with a vertical frequency.

For instance, a polynomial phase n → e(P (n)), where P : Z → R/Z

is a polynomial of degree ≤ s, is a degree ≤ s nilsequence with a vertical

frequency. Any nilsequence of degree ≤ s − 1 is trivially a nilsequence